C₁₉×#12296 #18018255

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12296

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

278 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27)(22,25,29,32)(23,26,30,33)(28,34,35,31)(37,43)(38,47)(39,50)(40,41)(44,58)(45,62)(46,57)(48,64)(49,54)(51,55)(52,53)(56,59)(60,67)(61,65);(20,22)(21,25)(23,28)(24,29)(26,31)(27,32)(30,35)(33,34)(36,37,42,46)(38,44,49,61)(39,51,53,62)(40,45,41,55)(43,52,57,50)(47,59,54,60)(48,65,64,58)(56,66,67,63);(20,23,24,30)(21,26,27,33)(22,28,29,35)(25,31,32,34)(36,38)(37,44)(39,48)(40,54)(41,47)(42,49)(43,56)(45,60)(46,61)(50,63)(51,65)(52,66)(53,64)(55,59)(57,67)(58,62)

Conjugacy classes & character table

589 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5284);